Lindemann-Hinshelwood

Here MX, Y designates an outer sphere or second sphere complex. There is every reason to suppose that formation and dissociation of MX, Y occurs at rates approaching the diffusional-control limit so that the slow conversion to MY is a negligible perturbation on the equilibrium of the first step. There is a similarity here the Langmuir, the Michaelis-Menten and the Lindemann-Hinshelwood schemes. 

In spite of the proper qualitative features of the Lindemann-Hinshelwood model, it does not correctly predict the much broader experimental fall-off behavior this is shown in Fig. 18, in which log(fe/fc ,) is plotted as a function of log(M = P/RT/Mj = Pc/RT). As evident from this figure, the actual rate at the center of fall-off (i.e., at PJ is depressed relative to the L-H model consequently, the transition of rate from low- to high-pressure limit occurs more gradually. 

Fig. 18. Comparison of the Lindemann-Hinshelwood model (dashed line) with experimental data (solid line).

In the preceding expression, log(FJ is related to the depression of the fall-off curve at the center relative to the L-H expression in a og k/k ) vs. log(2f/(l -I- X)) plot. The values for F<. can then be related to the properties of specific species and reaction and temperature using methods discussed in Gardiner and Troe (1984). In Fig. 19, values of F for a variety of hydrocarbon decompositions are presented. As evident from this figure, in the limit of zero or infinite temperatures and pressures, all reactions exhibit Lindemann-Hinshelwood behavior and F approaches unity. From this figure, it is clear that L-H analysis generally does an adequate job in... 

Of course, in a thermal reaction, molecules of the reactant do not all have the same energy, and so application of RRKM theory to the evaluation of the overall unimolecular rate constant, k m, requires that one specify the distribution of energies. This distribution is usually derived from the Lindemann-Hinshelwood model, in which molecules A become activated to vibrationally and rotationally excited states A by collision with some other molecules in the system, M. In this picture, collisions between M and A are assumed to transfer energy in the other direction, that is, returning A to A ... 

How thermal activation can take place following the Lindemann and the Lindemann-Hinshelwood mechanisms. An effective rate constant is found that shows the interplay between collision activation and unimolecular reaction. In the high-pressure limit, the effective rate constant approaches the microcanonical rate... 

More sophisticated treatments of Lindemann s scheme by Lindemann— Hinshelwood, Rice—Ramsperger—Kassel (RRK) and finally Rice— Ramsperger—Kassel—Marcus (RRKM) have essentially been aimed at re-interpreting rate coefficients of the Lindemann scheme. RRK(M) theories are extensively used for interpreting very-low-pressure pyrolysis experiments [62, 63]. 

In the Lindemann-Hinshelwood theory the Lindemann expression for the uni-molecular rate constant, Eq. (9), is still assumed to be correct, but an improved activation rate coefficient is obtained from the Hinshelwood formulation. The shape of the fall-off curve should therefore still be the simple form predicted by Lindemann. Reference to Fig. 2 shows that, for the cyclobutane decomposition reaction, the change in the activation rate coefficient brings the theory much closer to the experimental results, particularly at low pressure. However, the shape of the fall-off curve is still not correct the Lindemann-Hinshelwood model predicts a faU-off region that is too narrow, the true fall-off is broader. 

The second problem of the Lindemann-Hinshelwood theory is that the Lindemann plot is often far from a straight line, as can be seen by the example... 

Figure 2. Fall-off data for the dissociation of cyclobutane at 722 K [6,7], and comparison with Lindemann-Hinshelwood theory with 11 or 12 oscillators. The true number of vibrations is 30.

There is no reason for this expression to be consistent with the Lindemann straight line plot, but it is instructive to examine the physical reasons for the curvature. The low pressure limit is the same as in the Lindemann-Hinshelwood theory because the rate determining step is activation, which is dealt with in the same way in the two theories. This can be seen by taking the low pressure limit of Eq. (21). 

Figure 4. Proportional depletion by reaction with energy-dependent k2 E) and according to the Lindemann-Hinshelwood model (constant kz). The curve has been calculated at 999 K for ethane dissociation at the transition pressure.

In contrast, the Lindemann-Hinshelwood model assumes that all energized molecules react with the same rate constant k2- This model overestimates the contributions to uni from high energy states, and underestimates those from low-energy states. The true rate coefficient at some intermediate pressure will fall off faster than predicted by the Lindemann-Hinshelwood theory because of... 

The RRK theory has the virtue that it is very simple to apply and it does get close to the correct shape of the fall-off curve. As an example. Fig. 5 shows the fall-off curve calculated from classical RRK theory for the dissociation of cyclobutane using 14 oscillators. It can be seen that the theory is a considerable improvement on the Lindemann-Hinshelwood model. There are, however, some remaining problems. 

Troe proposed a similar approach to the calculation of the fall-off curve. In this case the zero-order approximation is the Lindemann-Hinshelwood model, formulated with the correct high and low-pressure limiting rate coefficients ... 

For the reactions of medium-sized molecules we have the following Lindemann-Hinshelwood theory, RRK theory. Slater s harmonic theory, RRKM theory, phase-space theory, absolute reaction rate theory, quasi-equilibrium theory, and several others. All of those are grouped under the umbrella of "transition state theory" (Robinson Holbrook, 1972 Forst, 1973). Among these theories, some are regarded as "inaccurate" or "outdated." But several rivals remain as viable alternatives on which to base a theoretical study of a reaction system, at least as far as Joiunal referees are concerned. 

Bimolecular steps involving identical species yield correspondingly simpler expressions. A3.4.8.2 THE LINDEMANN-HINSHELWOOD MECHANISM FOR UNIMOLECULAR REACTIONS... 

Figure A3.4.9. Pressure dependence of the effective unimolecular rate constant. Schematic fall-off curve for the Lindemann-Hinshelwood mechanism. A is the (constant) high-pressure limit of the effective rate constant...

This ensures the correct connection between the one-dimensional Kramers model in the regime of large friction and multidimensional unimolecular rate theory in that of low friction, where Kramers model is known to be incorrect as it is restricted to the energy diffusion limit. For low damping, equation (A3.6.29) reduces to the Lindemann-Hinshelwood expression, while in the case of very large damping, it attains the Smoluchowski limit... 

The expression for N t, E) in equation (A3.12.67) has been used to study [103.104] how the Porter-Thomas P k) affects the collision-averaged monoenergetic unimolecular rate constant k (Si, E) [105] and the Lindemann-Hinshelwood unimolecular rate constant T) [47]. The Porter-Thomas P k) makes k, E) pressure... 

Lindemann-Hinshelwood theory makes the assumption that a single collision with a bath gas molecule M is sufficient to deactivate AB to AB. In reality, each collision removes only a fraction of the energy. To account for the fact that not all collisions are fully deactivating, Troe (1983) developed a modification to the Lindemann-Hinshelwood rate expression. In the Troe theory, the right-hand side of... 

The Lindemann-Hinshelwood unimolecular rate constant ni( > E) is related to the lifetime distribution P t, E) according to (Slater, 1959 Bunker, 1964)... 

We have calculated the addition channel rate constant using the RRKM approach to unimolecular reaction rate theory, as formulated by Troe ( ) to match RRKM results with a simpler computational approach. The pressure dependence of the addition reaction (1) can be simply decribed by a Lindemann-Hinshelwood mechanism, written most conveniently in the direction of decomposition of the stable adduct ... 

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